Date post: | 10-Apr-2018 |
Category: |
Documents |
Upload: | sai-ashok-kumar-reddy |
View: | 218 times |
Download: | 0 times |
of 16
8/8/2019 Class II - Conservation Equations
1/16
Chapter 3
Conservation equations for Mass,
Momentum & Energy
8/8/2019 Class II - Conservation Equations
2/16
CONVECTION: Heat transfer process that occurs between a
solid surface and a fluid medium when they are at different
temperatures and have a relative motion between them.
W/m2 (Newtons law of cooling)
h influenced by thermo physical properties of fluid, flow velocity
and surface geometry.
Value varies from point to point as the properties vary with
temperature & location.
Local heat transfer coefficient & Average heat transfer coefficient
Determination of the value ofh is difficult (but critical)
Recap.
)( g!! TThA
Qq sconv
8/8/2019 Class II - Conservation Equations
3/16
Mechanism of Convection:
By pure conduction at the surface or boundary
Macroscopic fluid motion in the rest of the region
Macroscopic fluid movement enhances heat transfer, since it brings
cooler chunks of fluid into surface contact continuously, initiatinghigher rates of heat transfer
Recap.
0!x
x!
yy
Tkq
g
!
x
x
!TT
y
Tk
hs
y 0
Value of are known
Value of needs to be estimated
gTTk s
0!x
x
yy
T
8/8/2019 Class II - Conservation Equations
4/16
Boundary layer -proposed by Prandtl in 1904
Types:
Hydrodynamic or Velocity boundary layer
defined as that distance from the boundary in which the velocityreaches 0 to 99 % of the free stream velocity
Thermal boundary layer
defined as the distance from the boundary in which the temperaturedifference varies from 0 to 99 % of the initial temperature difference
Flow types Laminar & Turbulent flowcharacterized by Reynolds's number
Recap.
Q
V
K
lxxU
forceviscous
forceInertial!!!Re
For a flow over flat plate:
Re < 5 x 105 - Laminar flow
Re > 5 x 105 - Turbulent flow
8/8/2019 Class II - Conservation Equations
5/16
Boundary Layer ConceptPrandtl Number (Pr): defined as the ratio of the momentum diffusivity to
the thermal diffusivity.
E
K!!!
k
c
ydiffusivitThermal
ydiffusivitMomentum pPr
Prandtl Number (physically) is the ratio of kinematic viscosity () to the thermal
diffusivity ()
Kinematic viscosity indicates the impulse transport through molecular friction
whereas thermal diffusivity indicates the heat energy transport by conduction
process
Significance:provides a measure of relative effectiveness of the momentum and energy
transport by diffusion
connecting link between the velocity field and temperature field and its value
strongly influences relative growth of velocity and thermal boundary layers.
8/8/2019 Class II - Conservation Equations
6/16
Main purpose of convective heat transfer analysis is to
determine:
- heat transfer coefficient, h
How to solve a convection problem ?
Solve governing equations along with boundary conditions Governing equations include
1. conservation of mass
2. conservation of momentum
3. conservation of energy
Solving all these equations is a tiresome task.
Steady, two dimensional incompressible flow of constant property
Convection Analysis
8/8/2019 Class II - Conservation Equations
7/16
Convection Equations
Consider the parallel flow of a fluid over a surface
Assumptions:
laminar flow,
steady two-dimensional flow
Newtonian fluid
constant properties
The fluid flows over the surface with a uniform free-stream velocity V, but the
velocity within boundary layer is two-dimensional (u=u(x,y), v=v(x,y)).
Three fundamental laws
1. conservation of mass - continuity equation
2. conservation of momentum - momentum equation
3. conservation of energy - energy equation
8/8/2019 Class II - Conservation Equations
8/16
Conservation of mass principle the mass can not be created or destroyed during aprocess.
In a steady flow
Rate of mass flow into control volume = Rate of mass flow out of control volume
The mass flow rate is equal to: uA
Considering unit thickness,
x directionFluid enters the control volume from the left surface at a rate of u(dy.1)
Fluid leaves the control volume from the right surface at a rate of (dy.1)
Continuity Equation
x
x dx
x
uu
8/8/2019 Class II - Conservation Equations
9/16
Substituting the results in the conservation equation, we get
Simplifying and dividing by (dx dy), we get
1.1.1.1. dxdyy
vvdydx
x
uudxvdyu
x
x
x
x! VVVV
Continuity Equation
0!x
x
x
x
y
v
x
u (Continuity Equation in cartesian system)
Repeating the same procedure in y-direction, we get
0!x
x
x
x
z
v
r
v
r
vzrr (Continuity Equation in cylindrical system)
8/8/2019 Class II - Conservation Equations
10/16
Momentum Equation
The differential forms of the equations of motion in the velocity boundarylayer are obtained by applyingNewtons second law of motion to a differential
control volume element in the boundary layer.
Two type of forces:
body forces and surface forces.
Newtons second law of motion for a control volume is given by
Mass x (Acceleration in the specified direction)
= Net force (body & surface) acting in that direction
i.e
where the mass of the fluid element within the control volume is
dm = (dx.dy.1)
)()()(. xsurfacexbodyx FFa !x
8/8/2019 Class II - Conservation Equations
11/16
dyy
udx
x
udu
x
x
x
x!
Momentum EquationThe flow is steady and two-dimensional and thus u=u( x, y), the total
differential ofu is
Then the acceleration of the fluid element in the x direction becomes
The forces acting on a surface are due to pressure & viscous effects and areproportional to the area
Viscousstresscan be resolved into two perpendicular components Normalstress
Shearstress
Normal stress should not be confused with pressure
yuv
xuu
dt
dy
yu
dt
dx
xu
dt
duax
xx
xx!
xx
xx!!
8/8/2019 Class II - Conservation Equations
12/16
Momentum Equation
Neglecting the normal stresses, the netsurface force is given as
)1..(
)1..()1.()1.(
2
2
)(
dydxx
P
y
u
dydxx
P
ydydx
x
Pdxdy
yF xsurface
x
x
x
x!
x
x
x
x!
x
x
x
x!
Q
XX
The body forces is external force (like gravity) acting on the fluid particle andis proportional to the volume
If Bx is the body force per unit volume in the x direction, then body force in x-
direction is Bx.(dx.dy.1)
Substituting for mass, acceleration, surface force & body force (for x-direction) in Newton's law of motion, we get
8/8/2019 Class II - Conservation Equations
13/16
)1.()1..()1..( 2
2
dyxP
yudydxB
yuv
xuudydx x
xx
xx!
xx
xx QV
2
2
y
u
x
PB
y
uv
x
uu x
x
x
x
x!
x
x
x
xQV
Momentum Equation
Dividing by (dx dy)
x direction
2
2
x
u
y
PB
y
vv
x
vu y
x
x
x
x!
x
x
x
xQV y direction
The above sets of equation are known as Navier stokes equations for asteady, two dimensional flow of an incompressible, constant property
fluid
The above 2 equations along with continuity equation presents a set of 3equations to solve the three unknowns u, v and p.
8/8/2019 Class II - Conservation Equations
14/16
t
Tc
y
T
x
Tk p
x
x!
x
x
x
xV
2
2
2
2
x
x
x
x!
x
x
x
x
y
Tv
x
Tuc
y
T
x
Tk pV2
2
2
2
Energy EquationThe concept of conservation of energy has been already discussed in the
conduction chapter
We know that, for a two dimensional flow (without heat generation), theequation is given as
x
x
x
x!
x
x
y
Tv
x
Tu
t
T
But, the total time derivative of temperature is given as
Hence the energy balance equation is given as
8/8/2019 Class II - Conservation Equations
15/16
*
*
x
x
x
x!
x
x
x
xQV y
T
vx
T
ucy
T
x
T
k p2
2
2
2
Energy Equation
In certain cases (highly viscous fluid), certain work dW is done by fluid to
overcome the viscous effect which results in energy dissipation due to friction.
Accounting for the energy given out by viscous dissipation we have
222
2
-
x
x
x
x
-
x
x
x
x!*
x
v
y
u
y
v
x
u
Where is given as (after a lengthy analysis)
8/8/2019 Class II - Conservation Equations
16/16
Convective AnalysisTotal number of unknowns (Factors influencing the value ofh):
1.Three velocity components (u, v, w)2.Temperature (T)
3.Pressure (P)
4.Density ()
5.Viscosity ()
6.Thermal conductivity (k)
By assumption andsimplification, the number of unknowns are reduced as
1.Two velocity components (u, v)
2.Temperature (T)
3.Pressure (P)which requires a system of4 equations
The 4 equations are given by continuity equation, x momentum & y momentum equations and energy equation